Fill the big pot with salted water (at least 6 quarts with a tablespoon kosher salt), and heat to a rolling boil.,To make the sauce: Rinse the cherry tomatoes, basil, and mint leaves, and pat dry.,Put them all in the food-processor bowl along with the pine nuts, anchovies, garlic, salt, and peperoncino.,Process until pureed, scraping down the work bowl as needed.,With the machine running, pour in the olive oil in a slow, steady stream, forming a thick, emulsified sauce.,If serving now, scrape the sauce into a warmed deep serving bowl, and stir in the whole capers.,______,Cover the pot, return the water to a boil over high heat, then set the cover ajar, and cook the pasta until al dente.,While the pasta cooks, ladle some of the boiling water into the serving bowl to warm it.,Scrape in the sauce, and stir in the capers.,Lift the spaghetti from the pot, let drain for a moment, drop it into the bowl, and toss well, until all the strands are coated with sauce.,Sprinkle over it the parsley and grated cheese, and toss again.,Serve immediately, passing more grated cheese at the table.